Kendall Tau Correlation

Definition

• two series of data: $X$ and $Y$
• cooccurance of them: $(x_i, x_j)$, and we assume that $i<j$
• concordant: $x_i < x_j$ and $y_i < y_j$; $x_i > x_j$ and $y_i > y_j$; denoted as $C$
• discordant: $x_i < x_j$ and $y_i > y_j$; $x_i > x_j$ and $y_i < y_j$; denoted as $D$
• neither concordant nor discordant: whenever equal sign happens

Kendall’s tau is defined as

$$ \begin{equation} \tau = \frac{C- D}{\text{all possible pairs of comparison}} = \frac{C- D}{n^2/2 - n/2} \end{equation} $$